Vol. 3, No. 4, 2015

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ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
A crack with surface elasticity in finite plane elastostatics

Xu Wang and Peter Schiavone

Vol. 3 (2015), No. 4, 365–384
Abstract

We consider the effect of surface elasticity on a finite crack in a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. The surface mechanics is incorporated into the model of finite deformation by employing a version of the continuum-based surface/interface theory of Gurtin and Murdoch. A complete solution valid throughout the entire domain of interest is obtained by reducing the problem to two series of coupled Cauchy singular integrodifferential equations that are solved numerically using a collocation method. Our model predicts that, in general, the size-dependent Piola stresses exhibit a weak logarithmic singularity at the crack tips. For a crack in a special class of materials subjected to mode II loading, the stresses are bounded whereas the deformation gradients exhibit a logarithmic-type singularity at the crack tips.

Keywords
surface elasticity, hyperelastic material of harmonic type, crack, logarithmic singularity, Cauchy singular integrodifferential equations
Mathematical Subject Classification 2010
Primary: 30B99, 45E99, 74B20, 74R99
Milestones
Received: 10 May 2015
Revised: 20 August 2015
Accepted: 19 October 2015
Published: 12 February 2016

Communicated by Antonio Carcaterra
Authors
Xu Wang
School of Mechanical and Power Engineering
East China University of Science and Technology
130 Meilong Road
Shanghai 200237
China
Peter Schiavone
Department of Mechanical Engineering
University of Alberta
4-9 Mechanical Engineering Building
Edmonton, Alberta T6G 2G8
Canada